Self-similar measures and their Fourier transforms. II

A self-similar measure on R n was defined by Hutchinson to be a probability measure satisfying (formule...) here S j x = ρ j R j x+b j is a contractive similarity (0 < ρ j < 1, R j orthogonal) and the weights a j satisfy 0 < A j < 1, ∑ j=1 m a j = 1. By analogy, we define a self-similar distribution by the same identity *) but allowing the weights a j to be arbitrary complex numbers. We give necessary and sufficient conditions for the existence of a solution to (*) among distributions of compact support, and show that the space of such solutions is always finite dimensional. If F denotes the Fourier transformation of a self-similar distribution of compact support, let (formule...) where β is defined by the equation ∑ j=1 m ρ j −β |a j | 2 = 1